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In this work we prove sharp interpolatory estimates that exhibit a new link between Riesz transforms and directional projections of the Haar system in ${\mathbb{R}}^n.$ To a given direction $\epsilon \in \{0,1{\}}^n,\epsilon \ne (0,\dots ,0),$ we let $P^{(\epsilon )}$ be the orthogonal projection onto the span of those Haar functions that oscillate along the coordinates $\{i:{\epsilon }_i=1\}.$ When ${\epsilon }_{i_0}=1$ the identity operator and the Riesz transform $R_{i_0}$ provide a logarithmically convex estimate for the $L^p$ norm of $P^{(\epsilon )},$ see Theorem 1.1. Apart from its intrinsic interest Theorem 1.1 has direct applications to variational integrals, the theory of compensated compactness, Young measures, and to the relation between rank one and quasi convex functions. In particular we exploit our Theorem 1.1 in the course of proving a conjecture of L. Tartar on semi-continuity of separately convex integrands; see Theorem 1.5.